[AntonioLao]

One big question or one big mathematical truth taken for granted since its discovery is why Fourier analysis requires the appearance of imaginary unity? Searching for a logical explanation brings back question of its discoverer and the purpose of its discovery.

It was around 1807 when Fourier (1768-1830) started exact handling of infinite series, in particular, trigonometric series. He believed that in addition to Taylor series, these can also provide expansions of periodic as well as non-periodic functions. However, non-periodic functions are easier to work with if they are expanded by imaginary exponentials through the equivalence of Euler’s formula.

In 1811, he wrote Analytic Theory of Heat and submitted it for peer review by Lagrange, Laplace, and Legendre. For certain lack of rigor, it was rejected for publication even though a revised one won Fourier the grand prize of 1812 on the problem of heat conduction. Persistent research finally allowed Fourier to publish his treatise in 1822 incorporating his earlier papers intact.

A superficial investigation of an English translation of Chap. II, Sect. IX, Art. 162 showed that the time derivative of heat is equal to product difference of terms that with respect to the unit of length the dimension of 0, 1 for the unit of temperature, and -1 for the unit of time. Now, it is agreed that the square root of -1 is defined as the imaginary unity.

In the theory of matrices, a square matrix whose transpose is equal to its conjugate inverse is called a unitary matrix and a real unitary matrix is called an orthogonal matrix. Expressed as a matrix, the imaginary unity is a 2 by 2 skew-symmetric matrix with zero trace satisfying both unitarity and orthogonality and whose square is the negative identity matrix, I.